1. Field of the Invention
This invention is related to methods for collecting and processing nuclear magnetic resonance (NMR) transverse relaxation time (T.sub.2) spectra. In one application, the invention is related to methods for determining porosity measurement of a subterranean formation, or cores from such formation, using NMR tools or instruments.
2. Description of the Related Art
Nuclear magnetic resonance is used in the oil industry, among others, and particularly in certain oil well logging tools. NMR instruments may be used for determining, among other things, the fractional volume of pore space and the fractional volume of mobile fluid filling the pore space of earth formations. Methods of using NMR measurements for determining the functional volume of pore space and the fractional volume of mobile fluids are described, for example, in "Spin Echo Magnetic Resonance Logging: Porosity and Free Fluid Index Determination," M. N. Miller et al., Society of Petroleum Engineers paper no. 20561, Richardson, Tex., 1990. Further description is provided in U.S. Pat. No. 5,585,720, of Carl M. Edwards, issued Dec. 17, 1996, and assigned to Western Atlas International, Inc., entitled "Signal Processing Method For Multiexponentially Decaying Signals And Applications To Nuclear Magnetic Resonance Well Logging Tools." The disclosure of that patent is incorporated herein by reference.
Deriving accurate transverse relaxation time (T.sub.2) relaxation spectra from nuclear magnetic resonance (NMR) data from logging subterranean formations, or from cores from such formations, is critical to determining total and effective porosities, irreducible water saturations, and permeabilities of the formations. Accurate spectra are also essential to estimate T.sub.2 cutoff values and to obtain coefficients for the film model or Spectral Bulk Volume Irreducible (SBVI) model. Effective porosities are typically summations of partial porosities; however, distortion of partial porosity distributions has been commonly observed for a variety of reasons.
The most common NMR log acquisition and core measurement method employs T.sub.2 measurements using CPMG [Carr, Purcell, Meiboom and Gill] sequence, as taught by Meiboom and Gill in "Modified Spin-Echo Method for Measuring Nuclear Relaxation Time," Rev. Sci. Instrum. 1958, 29, pp. 688-691. In this method, the echo data in any given echo train are collected at a fixed time interval, the interecho time (TE). Usually, a few hundred to a few thousand echoes are acquired to sample relaxation decay.
Interecho time (TE), is one of the most important, controllable experimental parameters for CPMG measurements and can affect data interpretation. In logging operations using the MRIL.RTM. tool (made by Numar Corp.), TEs of 0.6 and 1.2 milliseconds (ms) are typically used to manipulate the relaxation decay data to include or exclude clay bound water (CBW) porosity. Not all clay bound water has a T.sub.2 relaxation time less than 4 ms and not all clay bound water may be resolved with a TE of 0.6 ms. For the purpose of this application, however, NMR T.sub.2 relaxation data with T.sub.2 .ltoreq.2.83 ms resolved with a TE of 0.6 ms will be referred to as CBW. It will be noted that other values of T.sub.2 and TE may be observed with other logging tools.
Both effective and total porosity, the latter defined as the sum of CBW and effective porosity, may be obtained with two CPMG measurements, as taught by Prammer et al in "Measurements of Clay-Bound Water and Total Porosity by Magnetic Resonance Logging," SPE paper 36522, presented at SPE Annual Tech. Conf. and Exhib., Denver, Co., 1996, pp. 311-320. This method implicitly assumes effective porosity, calculated from a TE of 1.2 ms acquisition, and effectively excludes the CBW porosity. This assumption is generally valid for rocks that contain little clay. However, with clay-rich rocks, a further correction may be required to improve the accuracy of the estimated effective porosity.
Interpretation of NMR core or log data is often started by inverting the time-domain CPMG echo decay into a T.sub.2 -parameter-domain distribution. In general, the T.sub.2 of fluids in porous rocks depends on the pore-size distribution and the type and number of fluids saturating the pore system. Because of the heterogeneous nature of porous media, T.sub.2 decays exhibit a multiexponential behavior. The basic equation describing the transverse relaxation of magnetization in fluid saturated porous media is ##EQU1## where M is magnetization, and effects of diffusion in the presence of a magnetic field gradient have not been taken into consideration and generally are negligible for short TE. In CPMG measurements, the magnetization decay is recorded (sampled) at a fixed period, TE; thus, a finite number of echoes are obtained at equally spaced time intervals, t=nTE, where n is the index for the n.sup.th echo, ##EQU2##
Equation (2) follows from Eq. (1) when noise and finite sampling are introduced. Because of the finite sampling of a continuous decay curve, information between the samples is not available. In order to estimate the unknown relaxation distribution function P(T.sub.2), a common approach is to use a set of predetermined relaxation times, T.sub.2, and to solve for the partial porosities, ppi, to fit the observed amplitudes, M(nTE). Using this approximation, the relaxation decay curve is modeled by the exponential equation ##EQU3## which expands to ##EQU4## where k is the index in the summation.
Mathematically, the multiexponential function is considered a valid approximation in that the function is linearly independent over distinct sampling points, as discussed by Hamming in Numerical Methods for Scientists and Engineers, 2.sup.nd Edition, McGraw-Hill, N.Y., 1973, pp. 617-619. This property guarantees that a unique and exact solution can be found provided that there is no noise and that a sufficient number of the fitting bins, T.sub.2k, are used to span all relaxation components in the underlying echo train. However, such strict conditions are not met in typical core and log NMR measurements. Consequently, the quality of the approximate solution is limited. Even small noise disturbances may substantially alter the solution. Generally, the presence of noise degrades the accuracy of T.sub.2 spectrum estimation for any T.sub.2 distribution patterns, but the short relaxation time components are the most affected. Furthermore, the individual exponential function, exp(-t/T.sub.2) is non-orthogonal. Thus, even without noise, depending on the fitting model bin selections, signals corresponding to short T.sub.2 components could be numerically determined as corresponding to other T.sub.2 components.
Inverting time domain NMR echo train data to T.sub.2 distributions with the multiexponential relaxation model of equation 4 above is known to be an ill-conditioned problem, particularly since signal-to-noise levels arc usually poor for NMR log data. The individual relaxation components in this model are not orthogonal to each other, which makes the estimation very vulnerable to noise. Often, a regularization technique is used to stabilize the results, which has a side effect of introducing artificial smoothing.
NMR relaxation decay is characterized with exponentials. Fourier transform (FT) of the time domain series can be used to analyze the frequency content of an exponential function. The frequency content of the multiexponential decay is the linear superposition of the FT of the individual exponentials. The FT of the continuous time-domain signal of a single exponential decay is ##EQU5## which is peaked at zero frequency and the tail depends on the decay constant T.sub.2. The discrete FT (DFT) of a single exponential decay echo ##EQU6## yields the frequency content and the spectrum density .vertline.X(k) .vertline..sup.2 /N, where N refers to length of the DFT and usually equals the number of data samples.
FIG. 1 shows the frequency contents and the spectral density of a single exponential decay with T.sub.2 =1 ms and several sampling times, TE=0.3, 0.6, 1.2, 2.4, and 4.8 ms, respectively. No noise is added. The number of echoes in each echo train is set to be different so that TE.sub.i.N.sub.i =512 msec for this simulation. Distortion of frequency content may occur with inadequate sampling.
FIG. 2 shows that long T.sub.2 components have more energy in the low frequency region. Short T.sub.2 components have relatively flat power spectra and lack distinctive frequency characteristics. Certain information is lost when T.sub.2 is not much larger than TE. Since the frequency contents of different exponential components overlap, the distortion of signal will affect several exponential components, not just the shortest one.
NMR effective porosity (MPHE) is often interpreted to be the summation of the capillary bound fluid as represented by the bulk volume or reducible (BVI) component and movable fluid as represented by the bulk volume movable (BVM) component but clay bound porosity (CBW) is not included. Conventionally NMR effective porosity (MPHE) is calculated as the summation of the partial porosities, i.e.: ##EQU7## where the summation in the equation includes all partial porosities with T.sub.2 .gtoreq.4 ms.
Generally, clay bound water (CBW) relaxes with T.sub.2 .ltoreq.2 ms. Since the standard NMR T.sub.2 log with, for example, the MRIL.RTM. tool of Numar Corp. uses the sampling period TE=1.2 msec, the CBW still can contribute to the NMR T.sub.2 log data. These signals often shift to adjacent, later T.sub.2 bins, contributing to inaccuracies in effective porosity and bulk-volume irreducible (BVI) estimates. Other tools may use different sampling periods, but the results would likely be similar.
In summary, noise, sampling rate, and the ill-conditioning of inversion and regularization contribute to smearing of the estimated T.sub.2 distribution and the shifting of the CBW signal to the higher T.sub.2 regions. This distortion is not easily rectified; even adding more bins with short T.sub.2 does not reduce the distortion of the T.sub.2 spectra. Our invention is a method for properly accounting for and correcting the CBW effect. The principles of the invention are also applicable in correcting other distortions of an NMR echo train.